David Malone wrote:
Terje Mathisen <[email protected]> writes:
One of the good points about Google's smear is the fact that they use a
half cosine to distribute the offset, which means that they have a time
function which is both continuous and monotonic, as well as having an
infinite number of defined derivatives, i.e. it is maximally smooth.
Doesn't it only have two smooth deritives at the end points or
[-w:w]? The usual function is constant 1 with all derivatves zero,
and so this is what the derivative should be at the endpoints. They
use (1.0 - cos(pi * t / w)) / 2.0, which is 1 at both end point,
has first derative zero, but the second deritive is -pi*pi/w/w.
The derivatives of sine/cosine are of course +/- cosine/sine, so they
stay smooth at all levels.
Google uses a half cosine, i.e. something like
adjustment = (1-cos(t * pi/adjustment_period))*adjustment_value/2;
Since the adjustment_value is +/- a second, the normal form is
adjustment = (1-cos(t*pi/adjustment_period))/2;
which is zero at t=0 and +1 at t==adjustment_period.
(It should be possible to stitch together something that is infinitely
smooth, probably using exp(-1/(x*x)), but it would requite a bit
more work.)
Doesn't seem to be needed?
Terje
--
- <Terje.Mathisen at tmsw.no>
"almost all programming can be viewed as an exercise in caching"
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